Kerr ergoregion

The Kerr ergoregion is the region where the asymptotically timelike stationary Killing field becomes spacelike. Its outer boundary is $r=M+\sqrt{M^2-a^2{\cos }^2\theta }$.

The Kerr realization of an ergosphere lies outside the outer horizon where the stationary Killing vector is spacelike; its outer boundary is $r=M+\sqrt{M^2-a^2{\cos }^2\theta }$.

For subextreme Kerr spacetime, the outer stationary-limit surface is $s=r-r_e(\theta )=0$, where $r_e=M+\sqrt{M^2-a^2{\cos }^2\theta }$. Away from the rotation axis, $r_e>r_{+}$ and $\Delta (r_e)=a^2{\sin }^2\theta >0$. Its normal is therefore spacelike and the surface timelike. Although the stationary Killing vector field has zero norm on it, it is not normal to it, and the surface is not a Killing horizon. The actual outer Killing horizon is $r=r_{+}$, generated by $K+{\Omega }_H\Phi$.

Superradiance is amplification of a scattered wave by extracting rotational or other available conserved energy from a background. For a rotating Kerr black hole, a bosonic mode of frequency $\omega$ and azimuthal integer $m$ can have negative horizon Killing-energy flux when $0<\omega <m{\Omega }_H$. The reflected energy can then exceed the incident energy. A stationary Killing flow need not define a globally positive particle energy.

Outside a subextremal Kerr black hole, regular causal curves obey $(-g_{t\varphi }-\sqrt{D})/g_{\varphi \varphi }\le d\varphi /dt\le (-g_{t\varphi }+\sqrt{D})/g_{\varphi \varphi }$, where $D=g_{t\varphi }^2-g_{tt}{g}_{\varphi \varphi }>0$. Positive radial and polar metric terms establish this necessary bound. In the open Kerr ergoregion both endpoints are positive for $a>0$, forcing co-rotation.

The Penrose process extracts rotational energy by allowing a fragment with negative stationary Killing energy to enter the black hole while another fragment escapes with more energy than the original body carried.

For a future-directed causal four-momentum $p$ crossing a future Killing horizon with generator $\xi =t+{\Omega }_H\phi$, the Lorentzian inner product obeys $p\cdot \xi \le 0$. With $E=-p\cdot t$ and $L=p\cdot \phi$, this is $E\ge {\Omega }_{H}L$. A massive particle has strict inequality. The same bound follows in the nonextremal quasistationary regime from the Physical-process first law for a rotating black hole and the second law of black-hole mechanics. Negative Killing energy in a Penrose process is allowed, but its angular momentum must satisfy this bound.